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%I #29 Jul 07 2024 06:40:00
%S 1,2,6,22,88,366,1556,6720,29396,129996,580276,2611290,11834116,
%T 53963190,247414100,1139860150,5274189156,24498929370,114199276476,
%U 534028437710,2504543749532,11777411979050,55518128412708,262301674637860,1241868060613788,5891050888101112,27995910970158108
%N Number of permutations of length n which avoid the patterns 4213 and 2143.
%H G. C. Greubel, <a href="/A165537/b165537.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..100 from David Bevan)
%H David Bevan, <a href="http://arxiv.org/abs/1510.06328">The permutation class Av(4213,2143)</a>, arXiv:1510.06328 [math.CO], 2015.
%H Kremer, Darla and Shiu, Wai Chee, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00042-6">Finite transition matrices for permutations avoiding pairs of length four patterns</a>, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Enumerations_of_specific_permutation_classes#Classes_avoiding_two_patterns_of_length_4">Permutation classes avoiding two patterns of length 4</a>.
%F G.f.: ((1-2*z) * (-1 + 5*z - 7*z^2 + 2*z^3 + (1-z)*sqrt(1 - 6*z + 5*z^2))) / (1 - 10*z + 24*z^2 - 20*z^3 + 4*z^4). - _David Bevan_, Sep 25 2015
%F Conjecture: n*a(n) + 2*(-9*n+7)*a(n-1) + (121*n-204)*a(n-2) + 28*(-14*n+37)*a(n-3) + 16*(42*n-151)*a(n-4) + 4*(-153*n+694)*a(n-5) + 4*(67*n-364)*a(n-6) + 40*(-n+6)*a(n-7) = 0. - _R. J. Mathar_, Jun 14 2016
%F a(n) ~ 12 * 5^(n + 3/2) / (121 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 07 2024
%e There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
%t Rest[CoefficientList[Series[((1-2*x)*(-1 +5*x -7*x^2 +2*x^3 +(1 - x)*Sqrt[1-6*x+5*x^2]))/(1-10*x+24*x^2-20*x^3+4*x^4), {x, 0, 50}], x]] (* _G. C. Greubel_, Oct 22 2018 *)
%o (PARI) z='z+O('z^66); Vec( ((1-2*z) * (-1 +5*z -7*z^2 +2*z^3 +(1-z) * sqrt(1 -6*z +5*z^2))) / (1 -10*z +24*z^2 -20*z^3 +4*z^4) ) \\ _Joerg Arndt_, Sep 27 2015
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-2*x)*(-1 +5*x -7*x^2 +2*x^3 +(1 - x)*Sqrt(1-6*x+5*x^2)))/(1-10*x +24*x^2-20*x^3+4*x^4))); // _G. C. Greubel_, Oct 22 2018
%K nonn
%O 1,2
%A _Vincent Vatter_, Sep 21 2009
%E More terms from _David Bevan_, Sep 25 2015
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